Properties

Label 2-3330-1.1-c1-0-4
Degree $2$
Conductor $3330$
Sign $1$
Analytic cond. $26.5901$
Root an. cond. $5.15656$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 5-s + 0.647·7-s − 8-s + 10-s − 3.68·11-s − 3.22·13-s − 0.647·14-s + 16-s − 1.35·17-s + 6.30·19-s − 20-s + 3.68·22-s − 5.22·23-s + 25-s + 3.22·26-s + 0.647·28-s + 3.87·29-s − 5.98·31-s − 32-s + 1.35·34-s − 0.647·35-s + 37-s − 6.30·38-s + 40-s + 0.242·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.447·5-s + 0.244·7-s − 0.353·8-s + 0.316·10-s − 1.11·11-s − 0.894·13-s − 0.172·14-s + 0.250·16-s − 0.328·17-s + 1.44·19-s − 0.223·20-s + 0.785·22-s − 1.08·23-s + 0.200·25-s + 0.632·26-s + 0.122·28-s + 0.718·29-s − 1.07·31-s − 0.176·32-s + 0.231·34-s − 0.109·35-s + 0.164·37-s − 1.02·38-s + 0.158·40-s + 0.0378·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3330\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 37\)
Sign: $1$
Analytic conductor: \(26.5901\)
Root analytic conductor: \(5.15656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3330} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3330,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8672046071\)
\(L(\frac12)\) \(\approx\) \(0.8672046071\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 + T \)
37 \( 1 - T \)
good7 \( 1 - 0.647T + 7T^{2} \)
11 \( 1 + 3.68T + 11T^{2} \)
13 \( 1 + 3.22T + 13T^{2} \)
17 \( 1 + 1.35T + 17T^{2} \)
19 \( 1 - 6.30T + 19T^{2} \)
23 \( 1 + 5.22T + 23T^{2} \)
29 \( 1 - 3.87T + 29T^{2} \)
31 \( 1 + 5.98T + 31T^{2} \)
41 \( 1 - 0.242T + 41T^{2} \)
43 \( 1 - 1.75T + 43T^{2} \)
47 \( 1 - 6.44T + 47T^{2} \)
53 \( 1 + 5.46T + 53T^{2} \)
59 \( 1 - 7.36T + 59T^{2} \)
61 \( 1 - 14.4T + 61T^{2} \)
67 \( 1 - 6.07T + 67T^{2} \)
71 \( 1 + 8.22T + 71T^{2} \)
73 \( 1 - 8.08T + 73T^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 - 2.14T + 83T^{2} \)
89 \( 1 - 5.81T + 89T^{2} \)
97 \( 1 + 4.20T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.502593387979124257499961992219, −7.81837034622502893680629227206, −7.45368648467053594403077233754, −6.60613733654781333446992001610, −5.51247721317037217039204524339, −4.96525021023895946376722601316, −3.84554899457031056581422792859, −2.83785287738202160502668448931, −2.02279419449629111608507366256, −0.59569632687008645347450801609, 0.59569632687008645347450801609, 2.02279419449629111608507366256, 2.83785287738202160502668448931, 3.84554899457031056581422792859, 4.96525021023895946376722601316, 5.51247721317037217039204524339, 6.60613733654781333446992001610, 7.45368648467053594403077233754, 7.81837034622502893680629227206, 8.502593387979124257499961992219

Graph of the $Z$-function along the critical line